Periodicity of Multidimensional Continued Fractions

Abstract

It is known that the continued fraction expansion of a real number is periodic if and only if the number is a quadratic irrational. In an attempt to generalize this phenomenon to other settings, Jun-Ichi Tamura and Shin-Ichi Yasutomi have developed a new algorithm for multidimensional continued fractions (Algebraic Jacobi-Perron algorithm) that involves cubic irrationals, and proved periodicity in some cubic number fields, such as $\mathbb{Q}(\sqrt[3]{m^3+1})$ where $m\in \mathbb{Z}$, and $\mathbb{Q}({\delta }_m)$ where ${\delta }_m$ is a root of $x^3-mx+1=0,m\in \mathbb{Z},m\ge 3$ with the algorithm. In this paper, we study some other types of number fields that give rise to periodic continued fractions using the Algebraic Jacobi-Perron algorithm obtaining results for $\mathbb{Q}(\sqrt[l]{m^l+1})$ for any positive integer $l$. Furthermore, we find that some families of cubic equations, such as $x^3+3a{x}^2+bx+ab-2a^3+1=0,b\le 3a^2-3,a,b\in \mathbb{Z}$, have roots that have periodic multidimensional continued fractions.

13 pages